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Exact Kahler Potential from Gauge Theory and Mirror Symmetry

Jaume Gomis, Sungjay Lee

TL;DR

This work proves that the exact partition function on the two-sphere for ${ m N}=(2,2)$ theories flowing to Calabi–Yau sigma models encodes the exact Kahler potential on the quantum Kahler moduli space, i.e., $Z( au_a,ar{ au}_a)=e^{- ext{K}( au_a,ar{ au}_a)}$, and uses this to access worldsheet instanton data. It provides two complementary proofs: (i) a localization-based analysis on a squashed sphere showing $Z_b$ is $b$-independent and reduces to a Ramond-sector ground-state overlap in the degenerate limit, (ii) a Landau–Ginzburg mirror analysis showing $Z= extstyle extstyle rac{}{}\

Abstract

We prove a recent conjecture that the partition function of N=(2, 2) gauge theories on the two-sphere which flow to Calabi-Yau sigma models in the infrared computes the exact Kahler potential on the quantum Kahler moduli space of the corresponding Calabi-Yau. This establishes the two-sphere partition function as a new method of computation of worldsheet instantons and Gromov-Witten invariants. We also calculate the exact two-sphere partition function for N=(2,2) Landau-Ginzburg models with an arbitrary twisted superpotential W. These results are used to demonstrate that arbitrary abelian gauge theories and their associated mirror Landau-Ginzburg models have identical two-sphere partition functions. We further show that the partition function of non-abelian gauge theories can be rewritten as the partition function of mirror Landau-Ginzburg models.

Exact Kahler Potential from Gauge Theory and Mirror Symmetry

TL;DR

This work proves that the exact partition function on the two-sphere for theories flowing to Calabi–Yau sigma models encodes the exact Kahler potential on the quantum Kahler moduli space, i.e., , and uses this to access worldsheet instanton data. It provides two complementary proofs: (i) a localization-based analysis on a squashed sphere showing is -independent and reduces to a Ramond-sector ground-state overlap in the degenerate limit, (ii) a Landau–Ginzburg mirror analysis showing $Z= extstyle extstyle rac{}{}\

Abstract

We prove a recent conjecture that the partition function of N=(2, 2) gauge theories on the two-sphere which flow to Calabi-Yau sigma models in the infrared computes the exact Kahler potential on the quantum Kahler moduli space of the corresponding Calabi-Yau. This establishes the two-sphere partition function as a new method of computation of worldsheet instantons and Gromov-Witten invariants. We also calculate the exact two-sphere partition function for N=(2,2) Landau-Ginzburg models with an arbitrary twisted superpotential W. These results are used to demonstrate that arbitrary abelian gauge theories and their associated mirror Landau-Ginzburg models have identical two-sphere partition functions. We further show that the partition function of non-abelian gauge theories can be rewritten as the partition function of mirror Landau-Ginzburg models.

Paper Structure

This paper contains 24 sections, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Supersymmetric Theories on Squashed Two-Sphere
  3. Superalgebra and Killing Spinors on Squashed Two-Sphere
  4. Supersymmetric Lagrangian on Squashed Two-Sphere
  5. A Proof of the Conjecture
  6. Exact Partition Function
  7. Degenerate Limit and Ground State Inner Product
  8. Mirror Symmetry and the $S^2$ Partition Function
  9. Supersymmetric Twisted Chiral Multiplets on S$^2$
  10. Landau-Ginzburg Proof of the Conjecture
  11. Mirror Symmetry and Hori-Vafa Conjecture
  12. Discussion
  13. Details of the Computation
  14. Matter Multiplet
  15. super multiplet
  16. ...and 9 more sections

Figures (1)

  • Figure 1: In the degenerate limit $b\to 0$, the two-sphere is deformed into a union of two infinitely stretched cigar-like geometries. The shaded region represents an infinitely long flat cylinder on which the standard $\cN=(2,2)$ supersymmetric theories in the Ramond sector are defined.